A hydrogen atom undergoes a transition from a $2p$ state to the $1s$ ground state. In the absence of a magnetic field, the energy of the photon emitted is $122$ nm. The atom is then placed in a strong magnetic field in the $z$-direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. How many different photon wavelengths are observed for the $2p\to 1s$ transition? What are the $m_l$ values for the initial and final states for the transition that leads to each photon wavelength?

Verified step by step guidance

1. Understand the Zeeman effect: When an atom is placed in a strong magnetic field, the energy levels of the atom split due to the interaction of the magnetic field with the orbital magnetic moment. This splitting depends on the magnetic quantum number (m_l).
2. Identify the initial and final states: The initial state is the 2p state, which has an orbital angular momentum quantum number l = 1. The magnetic quantum number m_l can take values -1, 0, and +1. The final state is the 1s state, which has l = 0 and m_l = 0 (no splitting occurs for the 1s state).
3. Determine the energy shifts: The energy shift due to the magnetic field is given by ΔE = m_lμ_BB, where μ_B is the Bohr magneton and B is the magnetic field strength. For the 2p state, the energy levels split into three sublevels corresponding to m_l = -1, 0, and +1.
4. Calculate the number of photon wavelengths: Transitions occur from each of the three 2p sublevels (m_l = -1, 0, +1) to the single 1s sublevel (m_l = 0). Since the energy difference for each transition is different due to the Zeeman splitting, three distinct photon wavelengths are observed.
5. List the m_l values for each transition: The transitions are (m_l = -1 → m_l = 0), (m_l = 0 → m_l = 0), and (m_l = +1 → m_l = 0). These correspond to the three different photon wavelengths observed in the presence of the magnetic field.

Key Concepts

Hydrogen Atom Energy Levels

The energy levels of a hydrogen atom are quantized, meaning electrons can only occupy specific energy states. The transition between these states, such as from 2p to 1s, results in the emission or absorption of photons with energies corresponding to the difference between these levels. The energy of the emitted photon can be calculated using the Rydberg formula, which relates the wavelengths of emitted light to the principal quantum numbers of the initial and final states.

Orbital Magnetic Moment

The orbital magnetic moment arises from the motion of electrons in their orbits around the nucleus. In the presence of a magnetic field, this moment interacts with the field, leading to phenomena such as the Zeeman effect, where energy levels split into multiple sub-levels. This splitting can result in the emission of photons with different wavelengths when transitions occur between these sub-levels, depending on the magnetic quantum number (ml) values.

Quantum Numbers and Selection Rules

Quantum numbers describe the properties of atomic orbitals and the electrons within them. For the hydrogen atom, the principal quantum number (n), azimuthal quantum number (l), and magnetic quantum number (ml) are crucial for determining allowed transitions. Selection rules dictate that transitions can only occur between states that differ in specific quantum numbers, influencing the number of distinct photon wavelengths observed during transitions, such as from 2p to 1s.